14 the cwcn term can be defined more explicitly by

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Unformatted text preview: e general expression for Qn can be analogously derived from equation 2 by incorporating all of the head-loss (13) This can be further simplified by dividing through by Qn to yield: . (14) The CWCn term can be defined more explicitly by substituting the right sides of equations 8 and 10 for the A and B terms in equation 14, resulting in .(15) The value of CWCn can be specified directly by the user (if LOSSTYPE is specified as “SPECIFYcwc” in dataset 2b), otherwise it will be calculated automatically by the model using equation 15 (for cases when LOSSTYPE is specified as “THIEM,” “SKIN,” or “GENERAL” in dataset 2b). The nature of the linear well-loss term indicates that, if the well is less than fully penetrating, the linear well-loss coefficient (B) would change inversely proportional to the fraction of penetration (α = ), and the cell-to-well conductance would be reduced accordingly. Also note that in an unconfined (convertible) cell, MODFLOW assumes that the saturated thickness changes as the water table rises or falls. Therefore, when the head changes in an unconfined cell containing a multi-node well, the value of CWCn must be updated by solving equation 15 again because both b and bw will have changed. Calculation of Water Level and Flow in the Multi-Node Well The basic numerical solution process is described by Bennett and others (1982), Halford and Hanson (2002), and Neville and Tonkin (2004). It consists of an iterative process with three basic steps: 1. solve the system of finite-difference equations for heads at each node of the grid, Conceptual Model and Numerical Implementation 2. solve for the composite head (water level) in the well, hWELL, and 3. solve for the flow rate at each node (Qn) and the net discharge from the well (Qnet). Specifying the Location of Multi-Node Wells The net flow to a multi-node well is simulated by summing the flow component to each node (Bennett and others, 1982; Fanchi and others, 1987; Halford and Hanson, 2002), which is defined by equation 12 and the common head in each node of the well. After the terms are collected and rearranged, the net flow rate between a multi-node well and the groundwater system is , 7 (16) where m is the total number of nodes in a multi-node well, and Qnet is the net flow between the well and the ground-water system (L3/T) and is equivalent to the flow at the wellhead (negative in sign for a discharge or withdrawal well). Because hWELL is common to all nodes in a multi-node well, equation 16 can be rewritten as (17) The value of hWELL is not known explicitly but is needed to estimate the flow rate between each well node and connected grid cell for a given multi-node well and to test that the drawdown does not exceed user-specified limits. Rearranging equation 17 gives the head in the well (Halford and Hanson, 2002): (18) In solving the governing ground-water flow equation, estimates of hWELL and Qn lag an iteration behind estimates of hn because equations 17 and 18 are solved explicitly assuming that hn is known. Halford and Hanson (2002) note that this causes slow convergence of the solver if the MNW cells are represented in MODFLOW as a general-head boundary (see McDonald and Harbaugh, 1988). Convergence is accelerated by alternately incorporating the MNW cells as specified-flux boundary conditions in odd iterations and as general-head boundaries in even iterations. That is, while solving the governing flow equation numerically, during odd numbered iterations the values of Qn and Qnet are specified from the latest known values and the head in the well is calculated using equation 18, and during even numbered iterations the head in the well is specified from the most recent known value and the values of Qn and Qnet are calculated using equations 12 and 17. In MODFLOW, a multi-node well must be linked to one or more nodes of the grid. The user has two options to accomplish this through the input datasets. One does not have to apply the same option to all wells. In the first approach, the user can specify the number of nodes of the grid that a particular well is associated with (by specifying the value of NNODES in dataset 2a). Then the layer-row-column location of each node (or cell) of the grid that is open to the multi-node well must be listed sequentially in dataset 2d. The order of the sequence must be from the first node that is located closest to the land surface or wellhead to the last node that is furthest from the wellhead. This ordering is required so that the model can properly route flow [and solute if the ground-water transport (GWT) process is active] through the borehole. There are no restrictions on locations of nodes, other than they be located in the active part of the grid. Thus, wells of any geometry—whether vertical, slanted, or horizontal—can be defined. The second approach is only applicable for vertical wells. It eliminates the need for the user to convert field data on depths or elevations of open intervals to corresponding layers of the grid. With this approach, the...
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